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Confining the state of light to a quantum manifold by engineered two-photon loss

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Science  20 Feb 2015:
Vol. 347, Issue 6224, pp. 853-857
DOI: 10.1126/science.aaa2085
  • Fig. 1 Schematic of the experiment.

    (A) Confinement of a quantum state belonging to a large Hilbert space into a 2D quantum manifold. The outer and inner cubes form a hypercube representing a multidimensional Hilbert space. The inner blue sphere represents the manifold of states spanned by the two coherent states Embedded Image. Quantum states such as the even and odd Schrödinger cat states Embedded Image also belong to this manifold, where Embedded Image is a normalization factor. Stabilizing forces direct all states toward the inner sphere without inducing any rotation in this subspace, as indicated by the purple arrows. (B) Two superconducting cavities are coupled through a Josephson junction. Pump and drive microwave tones are applied to the readout, creating the appropriate nonlinear interaction, which generates a coherent superposition of steady states in the storage. The readout output port is connected to an amplifier chain [(17), section 1.2]. Direct Wigner tomography of the storage is performed using its input port and the qubit mode. Q stands for “quality factor.” (C) Schematic spectrum of different modes involved in the experiment. The pump and drive tones are shown as vertical arrows. (D and E) Four-wave processes involved in the nonlinear damping and nonlinear drive, respectively, experienced by the storage. In (D), two photons of the storage combine and convert, stimulated by the pump tone, into a readout photon that is irreversibly radiated away by the transmission line. This process is balanced by the conversion of the drive tone, which, in the presence of the pump, creates two photons in the storage (E).

  • Fig. 2 Conversion of readout photons into pairs of storage photons.

    (A and B) CW spectroscopy of the readout in presence of the pump tone. The grayscale represents transmitted power of the probe tone through the readout as a function of probe frequency (horizontal axis) and pump frequency (vertical axis). In the top panel of (A), the usual Lorentzian response develops a sharp and deep dip, signaling conversion of probe photons into storage photons. The dip frequency ωdipp) decreases as the pump frequency increases. In the lower panel of (A), we plot Δf = ωdip/2π – (2ωs – ωp)/2π for each dip and see that the deviation of the data (open circles) to the theory (full line: Δf = 0) is only of the order of 0.24 MHz over a span of 20 MHz. The Stark-shifted value of ωs due to the pump is used [(17), fig. S5]. (B) Cut of the gray-scale map (A) at ωp/2π = 8.011 GHz. (C) Conversion seen from the storage, represented as the probability of not being in the vacuum state, as a function of drive frequency. The dashed and full lines in (B) and (C) are the result of a numerical computation of the steady-state density matrix of the system with Hamiltonian (Eq. 2) and loss operators Embedded Image, sweeping the drive frequency and keeping the pump frequency fixed. All parameters entering in theoretical predictions were measured or estimated independently. However, in (C) the theory was rescaled by a factor of 0.76 to fit the data. We believe that the need for this rescaling is a consequence of the unexplained modified qubit relaxation times when the pump and the drive are on [(17), section 1.4.9].

  • Fig. 3 Bistable behavior of the steady-state manifold of the nonlinearly driven damped storage oscillator.

    The central panel shows the theoretical classical equivalent of a potential of the storage nonlinear dynamics. The modulus of the velocity (color) has three zeroes corresponding to two SSSs Embedded Image and the saddle point Embedded Image. Trajectories initialized on the panel border converge to one of these two SSSs. These trajectories are curved due to the Kerr effect. The outside panels show the measured Wigner function W(α) of the storage after 10 μs of pumping for different initial states. For each panel, we initialize the storage in a coherent state of amplitude αk, where |αk| = 2.6 and arg(αk) is indicated in each panel. The storage converges to a combination of Embedded Image. The weight of each of these two states and the coherence of their superposition is set by the initial state. For the initial phases arg(αk) = 0, ±π/4, the storage mainly evolves to Embedded Image, with only a small weight on Embedded Image. On the other hand, for initial phases arg(αk) = ±3π/4, π, the state mainly evolves to Embedded Image, with a small weight on Embedded Image. For the initial phases arg(αk) = ±π/2, the initial state is almost symmetrically positioned with respect to the two states Embedded Image and has no definite parity (even and odd photon number states are almost equally populated). Hence, the state evolves to a mixture of Embedded Image. Re(α) and Im(α) denote the real and imaginary part of α, respectively.

  • Fig. 4 Time evolution of the storage state in the presence of the nonlinear drive and dissipation processes.

    The panels correspond to measured data (A), reconstructed density matrices (22) (B), and numerical simulations (C). They display the Wigner function after a pumping duration indicated at the top of each column. The storage is initialized in the quantum vacuum state at time t = 0 μs. First, the state squeezes in the Q quadrature (t = 2 μs). Small but visible negativities appearing at t = 7 μs indicate that the superposition of the SSSs shown in Fig. 3 is now coherent and that a continuous evolution from a squeezed state to a quantum state approximating a Schrödinger cat state is taking place. Finally, these negativities disappear as a consequence of the unavoidable storage photon loss, and the state decays into a statistical mixture of the two SSSs (t = 19 μs). (D) Storage photon number distribution P(n) measured using the photon number splitting of the qubit (29). At t = 2,7 μs, the n = 2 population is larger than n = 1. A similar population inversion is also present between n = 4 and n = 3 at t = 7 μs. The non-Poissonian character of the photon number distribution at t = 2,7 μs confirms the nonclassical nature of the dynamical states of the storage for these intermediate times.

Supplementary Materials

  • Confining the state of light to a quantum manifold by engineered two-photon loss

    Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge, M. Reagor, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Supplementary Text
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    • Tables S1 and S2
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